In the 18th century, the city of Konigsberg was
divided by a river. Seven bridges connected its
two banks and two islands. Residents wondered
whether it was possible to take a walk that crossed
every bridge exactly once. People tried many
routes. They could begin at different places
and they could cross the bridge in different
order. But every attempt failed. Then enter Leonard
Euler. Euler approached the problem differently.
He realized that the precise distances and directions
and shape of the bridges did not matter. He replaced
every region of land with a point and each bridge
with a line. The complicated map became a simple
network. Then Euler counted how many lines met
at each point. Every time a traveler enters a
point, another bridge is needed. So most points
in a successful route must have an even number
of bridges. Konigsberg had four points with an
odd number. So the desired walk was not possible.
Euler solved the problem by changing its representation.
He let go of the details that seemed essential.
and discovered a structure that had previously
been hidden. His insight helped begin what we
now call graph theory. It's a branch of math
that's used to study transportation, communication,
social networks, and a lot more. Now, Euler also
faced a profound change in his personal life.
His eyesight deteriorated over many years, and
he eventually would become nearly blind. And
yet, he continued to produce math at an extraordinary
rate. Now, it may be tempting to tell that this
is a simple story of heroic determination. Euler
overcame blindness because his mind was powerful
enough. But I think the fuller story is better.
Because Euler adapted. But he did not work alone.
He relied on his exceptional memory, yes, dictated
calculations, and received substantial help from
his sons and others, including colleagues and
assistants. His continued work was possible by
both personal perseverance and a community willing
to change how mathematics work would be done.
Now, Euler's blindness was not a gift, and we
do not need to pretend that losing his sight
made him a better mathematician. But Euler found
meaning not by declaring the loss good, but in
his response. He changed his methods. He accepted
help. And he continued to participate in the
work he valued. When I encounter a limitation,
sometimes I keep trying the same route with greater
force. Euler's mathematics suggests another possibility.
Perhaps the problem can be represented differently.
Or perhaps a tool or maybe a collaborator or
another method can make another route possible.
What limitation are you currently treating as
a command to stop? Could it instead be an invitation
to change the map? Or maybe just to ask for help?
Or maybe find another way across?